Average Error: 34.1 → 9.9
Time: 17.7s
Precision: 64
\[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]
\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}
↓
\begin{array}{l}
\mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\
\;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\

\mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\
\;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\

\mathbf{else}:\\
\;\;\;\;-1 \cdot \frac{c}{b}\\

\end{array}
double f(double a, double b, double c) {
        double r69066 = b;
        double r69067 = -r69066;
        double r69068 = r69066 * r69066;
        double r69069 = 4.0;
        double r69070 = a;
        double r69071 = c;
        double r69072 = r69070 * r69071;
        double r69073 = r69069 * r69072;
        double r69074 = r69068 - r69073;
        double r69075 = sqrt(r69074);
        double r69076 = r69067 + r69075;
        double r69077 = 2.0;
        double r69078 = r69077 * r69070;
        double r69079 = r69076 / r69078;
        return r69079;
}
↓
double f(double a, double b, double c) {
        double r69080 = b;
        double r69081 = -1.5501620157466267e+150;
        bool r69082 = r69080 <= r69081;
        double r69083 = 1.0;
        double r69084 = c;
        double r69085 = r69084 / r69080;
        double r69086 = a;
        double r69087 = r69080 / r69086;
        double r69088 = r69085 - r69087;
        double r69089 = r69083 * r69088;
        double r69090 = 1.611450844781215e-34;
        bool r69091 = r69080 <= r69090;
        double r69092 = 1.0;
        double r69093 = 2.0;
        double r69094 = r69093 * r69086;
        double r69095 = r69080 * r69080;
        double r69096 = 4.0;
        double r69097 = r69086 * r69084;
        double r69098 = r69096 * r69097;
        double r69099 = r69095 - r69098;
        double r69100 = sqrt(r69099);
        double r69101 = r69100 - r69080;
        double r69102 = r69094 / r69101;
        double r69103 = r69092 / r69102;
        double r69104 = -1.0;
        double r69105 = r69104 * r69085;
        double r69106 = r69091 ? r69103 : r69105;
        double r69107 = r69082 ? r69089 : r69106;
        return r69107;
}

Error

Bits error versus a

Bits error versus b

Bits error versus c

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original34.1
Target21.2
Herbie9.9
\[\begin{array}{l} \mathbf{if}\;b \lt 0.0:\\ \;\;\;\;\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{a \cdot \frac{\left(-b\right) - \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if b < -1.5501620157466267e+150

    1. Initial program 62.9

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around -inf 1.7

      \[\leadsto \color{blue}{1 \cdot \frac{c}{b} - 1 \cdot \frac{b}{a}}\]

    3. Simplified1.7

      \[\leadsto \color{blue}{1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)}\]

    if -1.5501620157466267e+150 < b < 1.611450844781215e-34

    1. Initial program 13.6

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]
    2. Using strategy rm

    3. Applied clear-num13.7

      \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}}}\]

    4. Simplified13.7

      \[\leadsto \frac{1}{\color{blue}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}}\]

    if 1.611450844781215e-34 < b

    1. Initial program 55.0

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)}}{2 \cdot a}\]

    2. Taylor expanded around inf 7.0

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification9.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.550162015746626746000974336574470460524 \cdot 10^{150}:\\ \;\;\;\;1 \cdot \left(\frac{c}{b} - \frac{b}{a}\right)\\ \mathbf{elif}\;b \le 1.61145084478121505718169973575148582501 \cdot 10^{-34}:\\ \;\;\;\;\frac{1}{\frac{2 \cdot a}{\sqrt{b \cdot b - 4 \cdot \left(a \cdot c\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{c}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (a b c)
  :name "quadp (p42, positive)"
  :precision binary64

  :herbie-target
  (if (< b 0.0) (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)) (/ c (* a (/ (- (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))))

  (/ (+ (- b) (sqrt (- (* b b) (* 4 (* a c))))) (* 2 a)))